Download thin conducting wires or cables interacting with a surrounding

Self-consistent Modeling of Thin Conducting Wires and their Interaction
with the Surrounding Electromagnetic Field
Göran Eriksson*1
1
ABB AB, Corporate Research
*Corresponding author: SE-721 78, Västerås, Sweden, [email protected]
Abstract: It is demonstrated how the RF module
can be used to approximately model thin
conducting wires or cables and how they interact
with a surrounding electromagnetic field. Despite
being non-stringent the method can reasonably
well predict currents induced by an applied
electromagnetic field in wires, and networks of
wires, as well as fields radiated from currentcarrying wires (antennas).
Keywords: Thin wires,
Radiated fields, Cables.
Induced
currents,
1. Introduction
In Finite Difference Time Domain (FDTD)
methods, where a regular structured mesh is used,
it is in fact possible to include the effect of
conducting wires having a radius much smaller
than the mesh size into a global 3D model [1]. A
Telegrapher's equation is employed to solve for
the wire current, thus allowing for wave
propagation and resonances. The trick is to take
into account the electromagnetic energy very
close to the wire surface by analytically including
it into the expressions for the wire inductance and
capacitance that appear in the Telegrapher´s
equations. Also the parallel component of the
global electric field is used to drive the wire
current. Thanks to the regular FDTD mesh
structure it is possible to describe the two-way
interaction with the surrounding electromagnetic
field in a fully self-consistent way. The FDTD
scheme has therefore become very popular for
modeling of all kinds of situations where thin
wires or cables are involved, such as wire
antennas and field coupling to and from cables
and cable networks. Note that this method allows
computation of both the field radiated from a
current-carrying wire and the current induced in
the cable due to an external time-varying
electromagnetic field.
Despite many attempts, see for example [2],
no similar technique being of practical use has
been presented for finite element methods (FEM).
Basically, the reason is the unstructured mesh into
which the wire is embedded and the mathematical
formulation of FEM. The proposed methods have
either required elements almost as small as the
wire radius and/or resulted in very complex
implementations. A consequence of this is that
FEM based solvers are not being considered
particularly efficient for solving problems related
to ElectroMagnetic Compatibility (EMC) issues
and wire antennas. EMC deals with problems
occurring due to cross-talk and interference to or
from conducting wires or cables; a typical
situation is illustrated by Fig. 1.
Here, we present an approximate method
which is very easy to implement and that can be
used to include wires having a radius smaller than
the typical element size into a 3D model. The
technique works for both low and high
frequencies. In version 4.3 of COMSOL
Multiphysics an implementation of the
Telegrapher's equation is introduced in the RF
module. Starting from this and adding the
appropriate couplings to the external fields we can
build up an approximately self-consistent model
of the wire and its surrounding environment. The
methodology is explained and illustrated with
some examples.
Figure 1. An example where electromagnetic
interference (EMI) between cables is very difficult to
model using conventional FEM techniques.
elements, one can derive the
expressions for these parameters:
2. The Telegrapher´s equations
In the Transmission Line Equation physics
node in COMSOL Multiphysics the frequency
domain equation for the voltage V along a twowire transmission line is given by
 1
V 

  G  iC V  0
x  R  iL x 
C  Cw 
(1)
Here, V(x) is the voltage between the two parallel
wires and x is the distance along the line. R, L, G,
and C are the resistance, inductance, conductance,
and capacitance per unit length of the line. The
line current I(x) is related to V(x) via
I
 1 V
R  iL x
(2)
It is important to note here that for the
transmission line it is assumed that the two wires
are so close to each other that all of the
corresponding electromagnetic energy is confined
to the region between the wires and that no
interaction takes place with the surroundings.
For the case of wires having radius a and axis
separation distance d, the field problem can be
solved analytically, resulting in the line
inductance and capacitance given by
L
C
 d 
ln  
 a

ln d a 
L  Lw 
(3)
(4)
3. Subcell model for a single thin wire
For a single wire far from other conductors it
is not possible to define and calculate line
parameters such as L and C describing all of the
electromagnetic energy associated with the wire
current and charge. It turns out, however, that one
can still use the Telegrapher´s equation (1), but
now with the line parameters describing only that
part of the energy which is localized in the
immediate vicinity of the wire surface, out to a
distance of the order of the mesh element size Δ.
In the FDTD formulation, with its regular
following
 
ln   (5)
2  2a 
2
ln  2a 
(6)
In order to include the driving force from
external sources and current in distant parts of the
wire, one adds the parallel component Ex of the
global electric field to the local field  V x in
(1):
 
1
 V


 E x   

x  R  iLw  x

(7)
G  iCw V  0
The current is now given by
I
1 
V 
 Ex 
 (8)
R  iL 
x 
In the finite element method, where the elements
are non-structured, it is not expected to have the
same degree of accuracy as in FDTD but, as we
will demonstrate, the agreement is acceptable for
many purposes. If one has the same element size
everywhere along the wire, this value is a good
approximation of Δ. Moreover we use a suitable
average value for the parallel electric field Ex. For
instance it is possible to employ the diskavg or
circavg operators.
Finally it should be noted that equation (7)
retains the form of equation (1), implying that
current conservation and continuity remain
natural boundary conditions.
4. Validation case I: A square-shaped loop
in an oscillating applied magnetic field
The first validation case consists of a planar
square-shaped wire loop immersed in an external
magnetic field. The method described above is
implemented using the electromagnetic wave
equation and transmission line physics nodes in
the RF module. Figure 2 shows the magnetic field
lines and the magnetic field amplitude in the plane
Figure 2. A conducting loop placed in a vertical
magnetic field. The amplitude of B is shown in the
plane of the loop (logarithmic color scale).
of the loop. It is clearly seen that the induced
current in the loop reduces the field amplitude
inside the loop, in accordance with Lenz´law.
Figure 3 illustrates the variation of V (green
curve), Eparallel (blue curve), and I (red curve)
along the loop. As expected, the voltage, i.e. the
wire charge density, varies but the current is
approximately constant around the loop.
However, the fluctuations in current indicate that
the convergence is relatively poor due to the nonstructured mesh and the associated difficulty to
define a proper Eparallel,
Figure 4. The induced loop current as function of
frequency and for three different values of the wire
radius a. Comparison between simulated and analytical
results
Since Lw and Cw depend on the wire radius a,
a series of simulations were performed, where the
loop current was calculated as function of
frequency for three different values of a. These
results were then compared to an analytical
expression derived using an exact formula for the
full loop inductance. As can be observed in Figure
4, the agreement is good. For low frequencies the
resistive part R of the loop impedance dominates,
resulting in a linear relation between I and f. As
the frequency increases the inductive part iωL
becomes dominant and the current is saturated.
Note that L consists of two parts: Lw and the part
represented by the extra term Ex in (7). If the latter
is omitted in (7), the current curves would still
saturate but on a much higher level. We have thus
shown that our model indeed gives a realistic
description.
5. Validation case II: A half wave dipole
antenna
Figure 3. Parallel electric field, wire voltage, and
current along the closed loop.
In the former validation case we have
demonstrated that our model is realistic for
describing the induction of wire currents due to
external fields. We now also want to verify that
we can model the electromagnetic field radiated
from a current-carrying wire.
To do this we look at one of the simplest
possible cases viz. a half wave dipole. This
vertical antenna consists of two rods, each 1 meter
long, connected via a short feed gap across which
a voltage is applied. Two models are compared:
Figure 7. 3D radiation pattern from the wire
dipole.
Figure 5. The two antenna models. Cylinder model
(left) and wire model (right).
one where the rods are resolved and modeled as
two perfectly conducting cylinders and one where
the antenna is modeled using two thin wires. The
two geometries are seen in Figure 5.
The current distribution in the wire model is
shown in Figure 6. Despite some fluctuations due
to convergence problems, the average current
profile is as expected and the radiation pattern is
almost identical to that of the resolved cylinder
model. This is confirmed in Figures 7 and 8. Also
the input impedance was calculated. It was found
to be 106 + 43i Ω for the cylinder model and 1014i Ω for the wire model.
The real parts, describing the radiated power,
obviously agree quite well. That the imaginary
parts, describing the inductive near-field energy,
differ significantly but this is not surprising since
the feed region is modeled very differently in the
two models.
Figure 8. Radiation pattern in a vertical plane.
Comparison between resolved cylinder and
transmission line wire antenna models.
6. Conclusions
Figure 6. Current distribution along the wire
dipole.
We have discussed and examined the
possibility to employ a technique borrowed from
the FDTD simulation method to model thin
conducting wires and how the interact with the
global electromagnetic field. Although the FEM
method cannot exactly model such thin wires, the
examples shown here demonstrate that our
method can be useful, provided the required
accuracy is not too high. In areas such as EMC,
where interference levels are measured in dB:s, it
may provide a means to calculate first estimates.
7. References
1. Taflove, A. and Hagness, S.C., Computational
Electrodynamics: The Finite-Difference TimeDomain Method, 3rd ed., Artech House, 2005.
2. Edelvik, F., Ledfelt, G., Lötstedt, P., and Riley,
D.J., “An unconditionally stable subcell model for
arbitrarily oriented thin wires in the FETD
method”, IEEE Trans. Antennas and Propagation,
Vol. 51, No. 8, August 2003, pp. 1797-1805.